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Chapter 9: Problem 7

Solve the given linear system. State whether the system is consistent, with independent or dependent equations, or whether it is inconsistent. $$ \left\\{\begin{array}{l} x-y=2 \\ x+y=1 \end{array}\right. $$

Short Answer

Expert verified
The system is consistent and independent with solution \( x = \frac{3}{2}, y = -\frac{1}{2} \).

Step by step solution

01

Write down the system of equations

The system of equations is given as: \( x - y = 2 \) and \( x + y = 1 \). We'll solve this system using the method of elimination.
02

Add the equations

Add the two equations together to eliminate \( y \):\[(x - y) + (x + y) = 2 + 1\]This simplifies to:\[2x = 3\].Solving for \( x \) gives \( x = \frac{3}{2} \).
03

Substitute to find y

Substitute \( x = \frac{3}{2} \) into the first equation \( x - y = 2 \):\[\frac{3}{2} - y = 2\]Solving for \( y \), we get:\[-y = 2 - \frac{3}{2}\]\[y = -\frac{1}{2}\]
04

Interpret the solution

The solution to the system is \( x = \frac{3}{2} \) and \( y = -\frac{1}{2} \). Since there is a unique solution, the system is consistent with independent equations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Elimination Method
The elimination method is a strategic approach used to solve a system of linear equations. It aims to eliminate one variable at a time, making it easier to solve for the remaining variable. Here's how it works:
  • Start by aligning your equations so that adding or subtracting them will cancel out one of the variables.
  • In the given problem, we had the equations:
    \( x - y = 2 \)
    \( x + y = 1 \).
  • By adding these equations, the \( y \) terms cancel each other out, leading to a simpler equation:
\[2x = 3\]
  • Once you have an equation with a single variable, solve it normally. Here, dividing both sides by 2 gives \( x = \frac{3}{2} \).
  • Substitute back to find the other variable. This method is effective when the coefficients of one variable are opposites or can be easily made into opposites through multiplication.
Consistent System
A consistent system of equations is one that has at least one solution. The solutions can either be a single unique solution or infinitely many solutions. What makes a system consistent is the compatibility of the equations.
  • In our example, after solving, we found the solution:
\[x = \frac{3}{2}, \quad y = -\frac{1}{2}\]
  • This means our system has a valid solution pair, thus proving that the system is consistent.
  • When systems are inconsistent, they involve contradictory equations that cannot all be true at the same time, leading to no solution.
Understanding whether a system is consistent helps in identifying the nature of the solution one should expect.
Independent Equations
Independent equations within a system indicate that no equation is a multiple of the other. This leads to a unique solution if the system is consistent.
  • The original equations, \( x - y = 2 \) and \( x + y = 1 \), are not multiples of each other. This means they are independent.
  • Independent equations graphically represent intersecting lines on a plane, with the point of intersection being the unique solution.
  • This is in contrast to dependent equations, which represent the same line and result in infinitely many solutions if consistent.
Independence of equations is a key characteristic that ensures there's enough information to find only one solution for all variables.
Unique Solution
The term "unique solution" refers to a single solution for the variables in a system of equations. It is a hallmark of a consistent system with independent equations.
  • The solution \( x = \frac{3}{2} \), \( y = -\frac{1}{2} \) suggests that there is exactly one set of values that satisfies both equations simultaneously.
  • A unique solution implies that the lines represented by the equations intersect at exactly one point.
  • This is different from systems with no solution (parallel lines) or with infinitely many solutions (overlapping lines).
In solving any linear system, discovering a unique solution helps validate the independence and consistency of the system.

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Most popular questions from this chapter

Solve the given linear system. State whether the system is consistent, with independent or dependent equations, or whether it is inconsistent. $$ \left\\{\begin{aligned} x-2 y+z-3 w &=0 \\ 8 x-8 y-z-5 w &=16 \\ -x-y+& 3 w=-6 \\ 4 x-7 y+3 z-10 w &=2 \end{aligned}\right. $$

Solve the given linear system. $$ \left\\{\begin{aligned} 2 x-3 y &=2 \\ x+2 y &=1 \\ 3 x+2 y &=-1 \end{aligned}\right. $$

Playing with Numbers The sum of three numbers is \(20 .\) The difference of the first two numbers is \(5,\) and the third number is 4 times the sum of the first two. Find the numbers.

Solve the given linear system. State whether the system is consistent, with independent or dependent equations, or whether it is inconsistent. $$ \left\\{\begin{aligned} x+y-5 z &=-1 \\ 4 x-y+3 z &=1 \\ 5 x-5 y+21 z &=5 \end{aligned}\right. $$

Find the partial fraction decomposition of the given rational expression. $$ \frac{x^{2}-x}{x(x+4)^{3}} $$
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